[...]... transport equation (Slattery, 1990) and it forms the basis for various mass transfer boundary conditions that apply at a phase interface In addition to the continuity equation and the jump condition, we need a set of N momentum equations to determine the species velocities, and we need chemical kinetic 8 Chemical Engineering constitutive equations for the homogeneous and heterogeneous reactions We also... by equation 1.36 and relations of the type given by equation 1.37, and this flux will be influenced by the convective, cA v , and diffusive, JA , fluxes The governing equations for cA and v are available to us in terms of equations 1.38 and 1.48, and here we consider the matter of determining JA To determine the mixedmode diffusive flux, we return to the Stefan–Maxwell equations and make use of equation... 1.36 and this requires knowledge of the concentration dependence of the homogeneous and heterogeneous reaction rates and information concerning the equilibrium adsorption isotherm In general, a solution of equation 1.69 for the system shown in Figure 1.7 requires upscaling from the point scale to the pore scale and this can be done by the method of volume averaging (Whitaker, 1999) 20 1.6 Chemical Engineering. .. purely intuitive; however, they are characteristic of the type of results obtained by careful analysis (Whitaker, 1986a; Quintard and Whitaker, 1994a–e; Whitaker, 1999) It is important to understand that Figures 1.1 and 1.2 The Art and Science of Upscaling 3 are not drawn to scale and thus are not consistent with the length scale constraints contained in equation 1.1 In order to determine the average rate... equation 1.32 is analytical and this level of analysis is necessary for an undergraduate course in reactor design The more practical problem deals with non-dilute solution diffusion and reaction in porous catalysts, and a rigorous analysis of that case is given in the following sections 1.4 Coupled, Non-linear Diffusion and Reaction Problems of isothermal mass transfer and reaction are best represented... make use of the mass average velocity and the mass diffusion velocity as indicated by equations 1.40 and 1.41, the molar flux in equation 1.35 takes the form c A vA = total molar flux cA v molar convective flux + c A uA mixed-mode diffusive flux (1.46) 16 Chemical Engineering Here we have decomposed the total molar flux into what we want, the molar convective flux, and what remains, i.e a mixed-mode diffusive... the macropore region and the micropore region are small enough, the concentrations of the reacting species can be treated as constants within this second averaging volume, and one can proceed to analyze the process of diffusion and reaction with Porous medium Porous catalyst Packed bed reactor I II Figure 1.3 Transport in a micropore–macropore model of a porous catalyst 4 Chemical Engineering a one-equation... in the process of diffusion and reaction We consider a two-phase system consisting of a fluid phase and a solid phase as illustrated in Figure 1.7 Here we have identified the fluid phase as the -phase and the solid phase as the -phase The foundations for the analysis of diffusion and reaction in this two-phase system consist of the species continuity equation in the -phase and the species jump condition... (Carbonell and Whitaker, 1984; Whitaker, 1986b) When homogeneous reactions can be ignored and the catalytic surface can be treated as quasisteady, the local volume-averaged transport equation simplifies to cA t accumulation + · cA vA transport = av RAs heterogeneous reaction and this result provides the basis for several special forms (1.84) The Art and Science of Upscaling 1.8 23 Convective and Diffusive... diffusion process in the pore may be transient (Carbonell and Whitaker, 1984; Whitaker, 1986b) Equations 1.13–1.17 represent the physical situation in the pore domain and we need equations that represent the physical situation in the porous medium domain This requires that we develop the area-averaged form of equation 1.13 and that we determine The Art and Science of Upscaling 11 under what circumstances .